M 



MACLAURIN'S Theorem. See Taylor's 7V 

 MAGNETIC Needle, variation and dip of.Sf-r Va> 

 MARS. See Planets, elements of. 

 MARS phases of. See Venus. 

 MAXIMA and Minima of quantities. 



1. To determine in what cases any quantity y, depending- upon .r, may 

 become a maximum or minimum, we must find the differential of the 

 equation which expresses the relation that they bear to each other, and 



make the quantity -~ = o. The resulting equation, combined with the 



original one, will give the values of x and y in which y is a maximum 

 or minimum. 



2. To determine when y is a maximum and when a minimum ; find 



rf 2 ?/ 



the value of -y- ^, and if it be negative, y is a maximum ; if it bf posi- 

 tive, a minimum. 



3. If -jt- and -j-^ both vanish, but -~ remain, then y will be neither 



a maximum or minimum at that place, but will pass through a point of 

 contrary flexure parallel to the abscissa. In like manner, if dy, d*y, 

 d s y vanish, but d* y remain, the ordinate y will be a maximum or mini- 

 num ; and if dy, d?y, <fiy, and d*y vanish, but $*y remain, it will pass 

 through a point of contrary flexure, and so on alternately. This follows 

 immediately from Taylor's theorem. 



4. If a quantity be a maximum or minimum, any power or root, mul- 

 tiple, or part, of the original quantity, will be a max. or min. 



Ex. 1. To divide a right line a into two parts, such that their rectan- 

 gle may be either a max. or min. 

 Here a x xt, =: maximum or minimum. Suppose^ = a x x*, then 



-^j- a 2 x = o ; :. x = ~. To find whether this solution gives a 



dy 

 max. or min., take the differential of the equation -^- -a2x; /, 



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